What B&W film you want to shoot depends a lot on who's going to process and print.
For easy lab processing I'd recommend Kodak T400CN or Ilford XP2. These are C41 films like your colour neg film and can be processed by any minilab.
If you want true B&W, then I'd recommend starting with Ilford HP5+ - it is very easy to process and very difficult to screw up, has a huge tonal range and a lovely grayscale. [I could never get good results from Tri-X

]. You could also try Delta 400 - I get true 400 speed from this film (with ID-11 1:1). If you want a slower film, there's the one and only Ilford PanF+.